LeviCivita , Permutations , InProduct , CrossProduct , ZeroVector , BaseVector , Identity , ZeroMatrix , DiagonalMatrix , IsMatrix , Normalize , Transpose , Determinant , Trace , Inverse , Minor , CoFactor , SolveMatrix , CharacteristicEquation , EigenValues , EigenVectors , IsHermitean , IsUnitary , SylvesterMatrix .

# Linear Algebra

This chapter describes the commands for doing linear algebra. They can be used to manipulate vectors, represented as lists, and matrices, represented as lists of lists.

LeviCivita | The totally anti-symmetric Levi-Civita symbol
Permutations | Form all permutations of a list
InProduct | Inner product of vectors
CrossProduct | Outer product of vectors
ZeroVector | Create a vector with all zeroes
BaseVector | Base vector
Identity | Identity matrix
ZeroMatrix | Matrix filled with zeroes
DiagonalMatrix | Construct a diagonal matrix
IsMatrix | Test whether argument is a matrix
Normalize | Normalize a vector
Transpose | Transpose of a matrix
Determinant | Determinant of a matrix
Trace | Trace of a matrix
Inverse | Inverse of a matrix
Minor | Principal minor of a matrix
CoFactor | Cofactor of a matrix
SolveMatrix | Solve a linear system
CharacteristicEquation | Characteristic polynomial of a matrix
EigenValues | Eigenvalues of a matrix
EigenVectors | Eigenvectors of a matrix
IsHermitean | Test whether a matrix is Hermitean
IsUnitary | Test whether a matrix is unitary
SylvesterMatrix | calculate the Sylvester matrix of two polynomials

###

LeviCivita -- The totally anti-symmetric Levi-Civita symbol

Standard math library

##### Calling format:

LeviCivita(list)

##### Parameters:

list - a list of integers 1 .. n in some order

##### Description:

"LeviCivita" implements the Levi-Civita symbol. This is generally useful for tensor calculus. `list` should be a list of integers, and this function returns 1 if the integers are in successive order, eg. `1,2,3,...` would return 1. Swapping two elements of this list would return -1. So, LeviCivita( `2,1,3` ) would evaluate to -1.

##### Examples:

    In> LeviCivita({1,2,3})
    Out> 1;
    In> LeviCivita({2,1,3})
    Out> -1;
    In> LeviCivita({2,2,3})
    Out> 0;

##### See also:

Permutations .

###

Permutations -- Form all permutations of a list

Standard math library

##### Calling format:

Permutations(list)

##### Parameters:

list - a list of elements

##### Description:

Permutations returns a list with all the permutations of the original list.

##### Examples:

    In> Permutations({a,b,c})
    Out> {{a,b,c},{a,c,b},{c,a,b},{b,a,c},
    {b,c,a},{c,b,a}};

##### See also:

LeviCivita .

###

InProduct -- Inner product of vectors

Standard math library

##### Calling format:

InProduct(a,b)

a . b (prec. 3)

##### Parameters:

a, b - vectors of equal length

##### Description:

The inner product of the two vectors "a" and "b" is returned. The vectors need to have the same size.

##### Examples:

    In> {a,b,c} . {d,e,f};
    Out> a*d+b*e+c*f;

##### See also:

CrossProduct .

###

CrossProduct -- Outer product of vectors

Standard math library

##### Calling format:

CrossProduct(a,b)

a X b (prec. 3)

##### Parameters:

a, b - three-dimensional vectors

##### Description:

The outer product (also called the cross product) of the vectors "a" and "b" is returned. The result is perpendicular to both "a" and "b" and its length is the product of the lengths of the vectors. Both "a" and "b" have to be three-dimensional.

##### Examples:

    In> {a,b,c} X {d,e,f};
    Out> {b*f-c*e,c*d-a*f,a*e-b*d};

##### See also:

InProduct .

###

ZeroVector -- Create a vector with all zeroes

Standard math library

##### Calling format:

ZeroVector(n)

##### Parameters:

n - length of the vector to return

##### Description:

This command returns a vector of length "n", filled with zeroes.

##### Examples:

    In> ZeroVector(4)
    Out> {0,0,0,0};

##### See also:

BaseVector , ZeroMatrix , IsZeroVector .

###

BaseVector -- Base vector

Standard math library

##### Calling format:

BaseVector(k, n)

##### Parameters:

k - index of the base vector to construct

n - dimension of the vector

##### Description:

This command returns the "k"-th base vector of dimension "n". This is a vector of length "n" with all zeroes except for the "k"-th entry, which contains a 1.

##### Examples:

    In> BaseVector(2,4)
    Out> {0,1,0,0};

##### See also:

ZeroVector , Identity .

###

Identity -- Identity matrix

Standard math library

##### Calling format:

Identity(n)

##### Parameters:

n - size of the matrix

##### Description:

This commands returns the identity matrix of size "n" by "n". This matrix has ones on the diagonal while the other entries are zero.

##### Examples:

    In> Identity(3)
    Out> {{1,0,0},{0,1,0},{0,0,1}};

##### See also:

BaseVector , ZeroMatrix , DiagonalMatrix .

###

ZeroMatrix -- Matrix filled with zeroes

Standard math library

##### Calling format:

ZeroMatrix(n, m)

##### Parameters:

n - number of rows

m - number of columns

##### Description:

This command returns a matrix with "n" rows and "m" columns, completely filled with zeroes.

##### Examples:

    In> ZeroMatrix(3,4)
    Out> {{0,0,0,0},{0,0,0,0},{0,0,0,0}};

##### See also:

ZeroVector , Identity .

###

DiagonalMatrix -- Construct a diagonal matrix

Standard math library

##### Calling format:

DiagonalMatrix(d)

##### Parameters:

d - list of values to put on the diagonal

##### Description:

This command constructs a diagonal matrix, that is a square matrix whose off-diagonal entries are all zero. The elements of the vector "d" are put on the diagonal.

##### Examples:

    In> DiagonalMatrix(1 .. 4)
    Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},
    {0,0,0,4}};

##### See also:

Identity , ZeroMatrix .

###

IsMatrix -- Test whether argument is a matrix

Standard math library

##### Calling format:

IsMatrix(M)

##### Parameters:

M - a mathematical object

##### Description:

`IsMatrix` returns `True` if `M` is a matrix, `False` otherwise. Something is considered to be a matrix if it is a list and all the entries of this list are themselves lists.

##### Examples:

    In> IsMatrix(ZeroMatrix(3,4))
    Out> True;
    In> IsMatrix(ZeroVector(4))
    Out> False;
    In> IsMatrix(3)
    Out> False;

##### See also:

IsVector .

###

Normalize -- Normalize a vector

Standard math library

##### Calling format:

Normalize(v)

##### Parameters:

v - a vector

##### Description:

Return the normalized vector of v: a vector going in the same direction but with length 1.

##### Examples:

    In> Normalize({3,4})
    Out> {3/5,4/5};
    In> % . %
    Out> 1;

##### See also:

InProduct , CrossProduct .

###

Transpose -- Transpose of a matrix

Standard math library

##### Calling format:

Transpose(M)

##### Parameters:

M - a matrix

##### Description:

Transpose returns the transpose of a matrix M. Because matrices are just lists of lists, this is a useful operation too for lists.

##### Examples:

    In> Transpose({{a,b}})
    Out> {{a},{b}};

###

Determinant -- Determinant of a matrix

Standard math library

##### Calling format:

Determinant(M)

##### Parameters:

M - a matrix

##### Description:

Returns the determinant of a matrix M.

##### Examples:

    In> DiagonalMatrix(1 .. 4)
    Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},
    {0,0,0,4}};
    In> Determinant(%)
    Out> 24;

###

Trace -- Trace of a matrix

Standard math library

##### Calling format:

Trace(M)

##### Parameters:

M - a matrix

##### Description:

Trace returns the trace of a matrix M (defined as the sum of the elements on the diagonal of the matrix).

##### Examples:

    In> DiagonalMatrix(1 .. 4)
    Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},
    {0,0,0,4}};
    In> Trace(%)
    Out> 10;

###

Inverse -- Inverse of a matrix

Standard math library

##### Calling format:

Inverse(M)

##### Parameters:

M - a matrix

##### Description:

Inverse returns the inverse of matrix M. The determinant of M should be non-zero. Because this function uses Determinant for calculating the inverse of a matrix, you can supply matrices with non-numeric matrix elements.

##### Examples:

    In> DiagonalMatrix({a,b,c})
    Out> {{a,0,0},{0,b,0},{0,0,c}};
    In> Inverse(%)
    Out> {{(b*c)/(a*b*c),0,0},{0,(a*c)/(a*b*c),0},
    {0,0,(a*b)/(a*b*c)}};
    In> Simplify(%)
    Out> {{1/a,0,0},{0,1/b,0},{0,0,1/c}};

###

Minor -- Principal minor of a matrix

Standard math library

##### Calling format:

Minor(M,i,j)

##### Parameters:

M - a matrix

i, j - positive integers

##### Description:

Minor returns the minor of a matrix around the element (i,j). The minor is the determinant of the matrix excluding the "i"-th row and "j"-th column.

##### Examples:

    In> A := {{1,2,3}, {4,5,6}, {7,8,9}};
    Out> {{1,2,3},{4,5,6},{7,8,9}};
    In> PrettyForm(A);

    /                    \
    | ( 1 ) ( 2 ) ( 3 )  |
    |                    |
    | ( 4 ) ( 5 ) ( 6 )  |
    |                    |
    | ( 7 ) ( 8 ) ( 9 )  |
    \                    /

    Out> True;
    In> Minor(A,1,2);
    Out> -6;
    In> Determinant({{2,3}, {8,9}});
    Out> -6;

##### See also:

CoFactor , Determinant , Inverse .

###

CoFactor -- Cofactor of a matrix

Standard math library

##### Calling format:

CoFactor(M,i,j)

##### Parameters:

M - a matrix

i, j - positive integers

##### Description:

CoFactor returns the cofactor of a matrix around the element (i,j). The cofactor is the minor times (-1)^(i+j).

##### Examples:

    In> A := {{1,2,3}, {4,5,6}, {7,8,9}};
    Out> {{1,2,3},{4,5,6},{7,8,9}};
    In> PrettyForm(A);

    /                    \
    | ( 1 ) ( 2 ) ( 3 )  |
    |                    |
    | ( 4 ) ( 5 ) ( 6 )  |
    |                    |
    | ( 7 ) ( 8 ) ( 9 )  |
    \                    /

    Out> True;
    In> CoFactor(A,1,2);
    Out> 6;
    In> Minor(A,1,2);
    Out> -6;
    In> Minor(A,1,2) * (-1)^(1+2);
    Out> 6;

##### See also:

Minor , Determinant , Inverse .

###

SolveMatrix -- Solve a linear system

Standard math library

##### Calling format:

SolveMatrix(M,v)

##### Parameters:

M - a matrix

v - a vector

##### Description:

SolveMatrix returns the vector x that satisfies the equation "M x = v". The determinant of M should be non-zero.

##### Examples:

    In> A := {{1,2}, {3,4}};
    Out> {{1,2},{3,4}};
    In> v := {5,6};
    Out> {5,6};
    In> x := SolveMatrix(A, v);
    Out> {-4,9/2};
    In> A * x;
    Out> {5,6};

##### See also:

Inverse , Solve , PSolve .

###

CharacteristicEquation -- Characteristic polynomial of a matrix

Standard math library

##### Calling format:

CharacteristicEquation(matrix,var)

##### Parameters:

matrix - a matrix

var - a free variable

##### Description:

CharacteristicEquation returns the characteristic equation of "matrix", using "var". The zeros of this equation are the eigenvalues of the matrix, Det(matrix-I*var);

##### Examples:

    In> DiagonalMatrix({a,b,c})
    Out> {{a,0,0},{0,b,0},{0,0,c}};
    In> CharacteristicEquation(%,x)
    Out> (a-x)*(b-x)*(c-x);
    In> Expand(%,x)
    Out> (b+a+c)*x^2-x^3-((b+a)*c+a*b)*x+a*b*c;

##### See also:

EigenValues , EigenVectors .

###

EigenValues -- Eigenvalues of a matrix

Standard math library

##### Calling format:

EigenValues(matrix)

##### Parameters:

matrix - a square matrix

##### Description:

EigenValues returns the eigenvalues of a matrix. The eigenvalues x of a matrix M are the numbers such that M*v=x*v for some vector.

It first determines the characteristic equation, and then factorizes this equation, returning the roots of the characteristic equation Det(matrix-x*identity).

##### Examples:

    In> M:={{1,2},{2,1}}
    Out> {{1,2},{2,1}};
    In> EigenValues(M)
    Out> {3,-1};

##### See also:

EigenVectors , CharacteristicEquation .

###

EigenVectors -- Eigenvectors of a matrix

Standard math library

##### Calling format:

EigenVectors(matrix,eigenvalues) Standard math library

##### Parameters:

matrix - a square matrix

eigenvalues - list of eigenvalues as returned by EigenValues

##### Description:

EigenVectors returns a list of the eigenvectors of a matrix. It uses the eigenvalues and the matrix to set up n equations with n unknowns for each eigenvalue, and then calls Solve to determine the values of each vector.

##### Examples:

    In> M:={{1,2},{2,1}}
    Out> {{1,2},{2,1}};
    In> e:=EigenValues(M)
    Out> {3,-1};
    In> EigenVectors(M,e)
    Out> {{-ki2/ -1,ki2},{-ki2,ki2}};

##### See also:

EigenValues , CharacteristicEquation .

###

IsHermitean -- Test whether a matrix is Hermitean

Standard math library

##### Calling format:

IsHermitean(A)

##### Parameters:

A - square matrix

##### Description:

IsHermitean(A) returns `True` if `A` is Hermitian and `False` otherwise. A is a Hermitian matrix iff Conjugate( Transpose A )= A. If A is a real matrix, it must be symmetric to be Hermitian.

##### Examples:

    In> IsHermitean({{0,I},{-I,0}})
    Out> True;
    In> IsHermitean({{0,I},{2,0}})
    Out> False;

##### See also:

IsUnitary .

###

IsUnitary -- Test whether a matrix is unitary

Standard math library

##### Calling format:

IsUnitary(A)

##### Parameters:

A - square matrix

##### Description:

This function tries to find out if A is unitary.

Matrix A is orthogonal iff A^(-1) = Transpose( Conjugate (A) ). This is equivalent to the fact that the columns of A build an orthonormal system (in respect to the scalar product defined by `InProduct`).

##### Examples:

    In> IsUnitary({{0,I},{-I,0}})
    Out> True;
    In> IsUnitary({{0,I},{2,0}})
    Out> False;

##### See also:

IsHermitean .

###

SylvesterMatrix -- calculate the Sylvester matrix of two polynomials

Standard math library

##### Calling format:

    SylvesterMatrix(poly1,poly2,variable)

##### Parameters:

`poly1` - polynomial

`poly2` - polynomial

`variable` - variable to express the matrix for

##### Description:

The function `SylvesterMatrix` calculates the Sylvester matrix for a pair of polynomials.

The Sylvester matrix is closely related to the resultant, which is defined as the determinant of the Sylvester matrix. Two polynomials share common roots only if the resultant is zero.

##### Examples:

    In> ex1:= x^2+2*x-a
    Out> x^2+2*x-a;
    In> ex2:= x^2+a*x-4
    Out> x^2+a*x-4;
    In> SylvesterMatrix(ex1,ex2,x)
    Out> {{1,2,-a,0},{0,1,2,-a},{1,a,-4,0},{0,1,a,-4}};
    In> Determinant(%)
    Out> 16-a^2*a- -8*a-4*a+a^2- -2*a^2-16-4*a;
    In> Simplify(%)
    Out> 3*a^2-a^3;

The above example shows that the two polynomials have common zeros if a=3.

##### See also:

Determinant , Simplify , Solve , PSolve .